Albert Madansky

The Fitting of Straight Lines when Both Variables are Subject to Error

Abstract Consider the situation where X and Y are related by Y = α + βX, where α and β are unknown and where we observe X and Y with error, i.e., we observe x = X + u and y = Y + v. Assume that Eu = Ev = 0 and that the errors (u and v) are uncorrelated with the true values (X and F). We survey and comment on the solutions to the problem of obtaining consistent estimates of α and β from a sample of (x, y)s, (1) when one makes various assumptions about properties of the errors and the true values other than those mentioned above, and (2) when one has various kinds of “additional information” which aids in constructing these consistent estimates. The problems of obtaining confidence intervals for β and of testing hypotheses about β are not discussed, though approximate variances of some of the estimates of β are given. * This paper is an outgrowth of a Masters Thesis submitted to the Department of Statistics, University of Chicago. I am indebted for helpful comments and criticisms to T. E. Harris, W. H. Kr...

APPROXIMATE CONFIDENCE LIMITS FOR THE RELIABILITY OF SERIES AND PARALLEL SYSTEMS

Suppose a complex mechanism, e.g., a missile, is built up from a number of different types of components, where the reliability of each of the components has been estimated by means of separate tests on each of the components. This paper gives a method for combining such data to determine approximate confidence limits for the reliability of the complete mechanism. More precisely, a method of determining approximate confidence limits for the reliability of “series,” “parallel,” and “seriesparallel” systems is given, based on observed failures of the individual components. It is assumed that the failures are independent, and that failures of a given component follow a binomial distribution with unknown parameter, the component reliability. The large-sample properties of the likelihood-ratio test are then used to construct the appropriate confidence limits for the system reliability.

Least squares estimation in finite Markov processes

The usual least squares estimate of the transitional probability matrix of a finite Markov process is given for the case in which, for each point in time, only the proportions of the sample in each state are known. The purpose of this paper is to give another estimate of this matrix and to investigate the properties of this estimate. It is shown that this estimate is consistent and asymptotically more efficient than the previously considered estimate in a sense defined in this paper.

INSTRUMENTAL VARIABLES IN FACTOR ANALYSIS

The factor analysis model is rewritten as a system of linear structural relations with errors in variables. The method of instrumental variables is applied to this revised form of the model to obtain estimates of the factor loading matrix. The relation between this method and interbattery analysis, proportional profile analysis, and canonical factor analysis is pointed out. In addition, an estimation procedure based on replicated sampling different from proportional profile analysis is given.

Tests of Homogeneity for Correlated Samples

Abstract This paper presents techniques for testing various hypotheses related to the notion of temporal homogeneity of a population each of whose members can belong to any one of S states at any time. These hypotheses include Cochrans interchangeability hypothesis, the hypothesis of strict exchange, the usual homogeneity hypothesis for multinomial distributions in the presence of correlated samples, and the hypothesis that a first order Markov chain is in a steady state.

Statistical Estimation Procedures for the “Burn-In” Process

Thii study deals with the “burn-in” process. In this process, items whose failure rate is assumed to decrease with time are put on test (burnt-in) until a fixed amount of time has elapsed (truncated sampling) or until a fixed number of failures have occurred (censored sampling). The purpose is to identify and eliminate poor-quality or defective items. For both of the modes of observation described, we provide a conservative upper confidence bound for the failure rate at the time the burn-in process ends, and the maximum likelihood estimate (MLE) of the failure-rate function. These results are valid under quite general conditions. In particular, we do not require that the form of the life distribution be known. The MLE is obtained under the sole assumption that the failure-rate function is decreasing. The confidence bound is obtained under the even weaker assumption that the failure rate at the time observation ends is no larger than the failure rate throughout the period of observation. For the truncated ...

On Conjoint Analysis and Quantal Choice Models

Conjoint analysis (or trade-off analysis) has become increasingly used in marketing. Essentially, the techniques attempt to parse the appeals of a product or concept into a set of factors, assign utility values to the separate levels of each factor, and finally, under the assumption of separability, determine the utility value of any product or concept by adding (or multiplying) the utility values of the individual levels of each of the factors embodied in the product or concept. A good expository description of conjoint analysis is given in Green and Srinivasan (1978). For notational convenience, I present a brief description herein. Formally, if we are considering n factors, with the ith factor (i = 1, . . . , n) having mi levels, then we shall represent a bundle as an n-tuple (i1, i2, . . , ia), where

SPURIOUS CORRELATION DUE TO DEFLATING VARIABLES

Abstract : This Memorandum shows that when a homogeneous linear regression of a normally distributed variable Y on two nromally distributed variables X and Z is deflated by Z, then when X and Y are uncorrelated the deflated dependent variable Y/Z and independent variable X/Z are either uncorrelated or perfectly correlated. Thus, existing approximations to the covariance of these deflated variables are poor. A new approximation to this covariance is given which has the same defect for normally distributed variables, but which could otherwise be better than existing ones. (Author)

DETERMINANTAL METHODS IN LATENT CLASS ANALYSIS

Some extensions of the existing determinantal methods for solving the accounting equations in latent class analysis are presented. These extensions cover more cases than previous methods, give rise to new sufficient conditions for identifiability of the latent class model, and give insight into the necessity of various sufficient conditions for identifiability. These implications to the identifiability problem are discussed.

Foundations of Econometrics.

Advanced Textbooks in Economics, Volume 7: Foundations of Econometrics focuses on the principles, processes, methodologies, and approaches involved in the study of econometrics. The publication examines matrix theory and multivariate statistical analysis. Discussions focus on the maximum likelihood estimation of multivariate normal distribution parameters, point estimation theory, multivariate normal distribution, multivariate probability distributions, Euclidean spaces and linear transformations, orthogonal transformations and symmetric matrices, and determinants. The manuscript then ponders on linear expected value models and simultaneous equation estimation. Topics include random exogenous variables, maximum likelihood estimation of a single equation, identification of a single equation, linear stochastic difference equations, and errors-in-variables models. The book takes a look at a prolegomenon to econometric model building, tests of hypotheses in econometric models, multivariate statistical analysis, and simultaneous equation estimation. Concerns include maximum likelihood estimation of a single equation, tests of linear hypotheses, testing for independence, and causality in economic models. The publication is a valuable source of data for economists and researchers interested in the foundations of econometrics.